In how many ways can the letters of the word ABACUS

lheiannie07 wrote:In how many ways can the letters of the word ABACUS be rearranged such that the vowels always appear together?

A. 6!/2!
B. 3!*3!
C. 4!/2!
D. 4! *3!/2!
E. 3!*3!/2

Can some experts show me the best solution in this problem?

OA D

Permutation of N objects that have N1 identical objects of type 1 and N2 identical objects of type 2 and Nk identical objects of type k is

N!/(N1!*N2!..Nk!)

We have 3 vowels and 2 of then are identical (2 As), so the number of ways to arrange them are

3!/2!

Since we have to keep the vowels together, let us consider then as 1 group and call it X.

We then have to arrange the following alphabets.
XBCS

These can be arranged in 4! ways.

Hence the total number of ways in which these alphabets can be arranged is

4!*3!/2!

Hence D.